Rotational spectroscopy selection rules

The methyl iodide molecule is studied using microwave (pure rotational) spectroscopy. The following integral governs the rotational selection rules for transitions labeled J, M, K... 

Beyond such electronic symmetry analysis, it is also possible to derive vibrational and rotational selection rules for electronic transitions that are El allowed. As was done in the vibrational spectroscopy case, it is conventional to expand i j (R) in a power series about the equilibrium geometry of the initial electronic state (since this geometry is more characteristic of the molecular structure prior to photon absorption) ... 

This spectrum is called a Raman spectrum and corresponds to the vibrational or rotational changes in the molecule. The selection rules for Raman activity are different from those for i.r. activity and the two types of spectroscopy are complementary in the study of molecular structure. Modern Raman spectrometers use lasers for excitation. In the resonance Raman effect excitation at a frequency corresponding to electronic absorption causes great enhancement of the Raman spectrum. 

Finally, for the determination of selection rules for rotational spectroscopy it is necessary to find the wavefimcdons for this problem. This subject will be left for further development as given in numerous texts on molecular spectroscopy. 

The dipole and polarization selection rules of microwave and infrared spectroscopy place a restriction on the utility of these techniques in the study of molecular structure. However, there are complementary techniques that can be used to obtain rotational and vibrational spectrum for many other molecules as well. The most useful is Raman spectroscopy. 

Both infrared and Raman spectra are concerned with measuring molecular vibration and rotational energy changes. However, the selection rules for Raman spectroscopy are very different from those of infrared - a change of polarisability... 

Recall that homonuclear diatomic molecules have no vibration-rotation or pure-rotation spectra due to the vanishing of the permanent electric dipole moment. For electronic transitions, the transition-moment integral (7.4) does not involve the dipole moment d hence electric-dipole electronic transitions are allowed for homonuclear diatomic molecules, subject to the above selection rules, of course. [The electric dipole moment d is given by (1.289), and should be distinguished from the electric dipole-moment operator d, which is given by (1.286).] Analysis of the vibrational and rotational structure of an electronic transition in a homonuclear diatomic molecule allows the determination of the vibrational and rotational constants of the electronic states involved, which is information that cannot be provided by IR or microwave spectroscopy. (Raman spectroscopy can also furnish information on the constants of the ground electronic state of a homonuclear diatomic molecule.)... 

Spectroscopy is concerned with the observation of transitions between stationary states of a system, with the accompanying absorption or emission of electromagnetic radiation. In this section we consider the theory of transition probabilities, using time-dependent perturbation theory, and the selection rules for transitions, particularly those relevant for rotational spectroscopy. 

In many cases, the infrared and Raman rotation-vibration spectra contribute complementary structure data, particularly for highly symmetric molecules. Due to the significantly different selection rules a greater line density is observed for Raman due to a larger selection of allowed changes in the rotational energy compared to infrared gas spectra. Raman spectroscopy is, on these grounds, also a valuable supplement to infrared studies. 

Rotational features of almost aU H-bonded complexes in the gaseous phase appear in the microwave region, with wavenumbers less than 10 cm They correspond to transitions between pure rotational levels, pure meaning that vibrations remain unchanged, or no vibrational transition accompanies such rotational transitions. Rotational features, however, also appear in the IR spectra of these H-bonded complexes. IR bands correspond to transitions between various vibrational levels of a molecule. When this molecule is isolated, as in the gas phase, these transitions are always accompanied by transitions between rotational levels that obey the same selection rules as pure rotational transitions detected in microwave spectroscopy. The information conveyed by these rotational features in IR spectra are therefore most similar to those conveyed by microwave spectra, even if the mechanism at the origin of their appearance is different. Their interests lie in the use of an IR spectrometer, a common instrument in many laboratories, instead of a microwave spectrometer, which is a much more specialized instrament. However, the resolution of usual IR spectrometers are lower than that of microwave spectrometers that use Fabry-Perot cavities. This IR technique has been used in the case of simple H-bonded dimers with relatively small moments of inertia, such as, for instance, F-H- -N C-H (3). Such complexes are far from simple to manipulate, but provide particularly simple IR spectra with a limited number of bands that do not show any overlap. 

Pure rotational transitions of symmetrical diatomic molecules like dihydrogen are forbidden in infrared spectroscopy by the dipole selection rule but are active in Raman spectroscopy because they are anisotropically polarisable. They are in principle observable in INS although the scattering is weak except for dihydrogen. These rotational transitions offer the prospect of probing the local environment of the dihydrogen molecule, as we shall see in this chapter. 

The symmetry of an isolated atom is that of the full rotation group R+ (3), whose irreducible representations (IRs) are D where j is an integer or half an odd integer. An application of the fundamental matrix element theorem [22] tells that the matrix element (5.1) is non-zero only if the IR DW of Wi is included in the direct product x of the IRs of ra and < f. The components of the electric dipole transform like the components of a polar vector, under the IR l)(V) of R+(3). Thus, when the initial and final atomic states are characterized by angular momenta Ji and J2, respectively, the electric dipole matrix element (5.1) is non-zero only if D(Jl) is contained in Dx D(j 2 ) = D(J2+1) + T)(J2) + )(J2-i) for j2 > 1 This condition is met for = J2 + 1, J2, or J2 — 1. However, it can be seen that a transition between two states with the same value of J is allowed only for J 0 as DW x D= D( D(°) is the unit IR of R+(3)). For a hydrogen-like centre, when an atomic state is defined by an orbital quantum number , this can be reduced to the Laporte selection rule A = 1. This is of course formal, as it will be shown that an impurity state is the weighted sum of different atomic-like states with different values of but with the same parity P = ( —1) These states are represented by an atomic spectroscopy notation, with lower case letters for the values of (0, 1, 2, 3, 4, 5, etc. correspond to s, p, d, f, g, h, etc.). The impurity states with P = 1 and -1 are called even- and odd-parity states, respectively. For the one-valley EM donor states, this quasi-atomic selection rule determines that the parity-allowed transitions from Is states are towards np (n > 2), n/ (n > 4), nh (n > 6), or nj (n > 8) states. For the acceptor states in cubic semiconductors, the even- and odd-parity states labelled by the double IRs T of Oh or Td are indexed by + or respectively, and the parity-allowed transition take place between Ti+ and... 

Electrons, protons and neutrons and all other particles that have 5 = are known as fermions. Other particles are restricted to 5 = 0 or 1 and are known as bosons. There are thus profound differences in the quantum-mechanical properties of fermions and bosons, which have important implications in fields ranging from statistical mechanics to spectroscopic selection rules. It can be shown that the spin quantum number S associated with an even number of fermions must be integral, while that for an odd number of them must be half-integral. The resulting composite particles behave collectively like bosons and fermions, respectively, so the wavefunction symmetry properties associated with bosons can be relevant in chemical physics. One prominent example is the treatment of nuclei, which are typically considered as composite particles rather than interacting protons and neutrons. Nuclei with even atomic number therefore behave like individual bosons and those with odd atomic number as fermions, a distinction that plays an important role in rotational spectroscopy of polyatomic molecules. 

Infrared and Raman spectroscopy are important analytical tools used to investigate a wide variety of molecules in the solid, liquid, and gas states, and yielding complementary information about molecular structure and molecular bonds. Both methods supply information about resonances caused by vibration, vibration-rotation, or rotation of the molecular framework, but because the interaction mechanism between radiation and the molecule differs in the two types, the quantum-mechanical selection rules differ as well. Therefore, not all of the molecular motions recorded by one type of spectroscopy will necessarily be recorded by the other. The geometrical configuration of the molecule and the distribution of electrical charge within that configuration determine which molecular motions may appear in each type of spectrum. 

The complexity of Raman spectra for polymers is reduced as with infrared spectra because vibrations of the same type superimpose. In addition, as with infrared spectroscopy, selection rules aid in determining which molecular vibrations are active. However, the criterion for Raman aetivity is a change in bond polarizability with molecular vibration or rotation in contrast to the infrared criterion of a change in dipole moment (Figure 6.6). This means that, for molecules such as carbon dioxide that show both a change in dipole moment and a change in polarizability,... 

D13.3 (I) Rotational Raman spectroscopy. The gross selection rule is that the molecule mustbe anisotropically polarizable, which is to say that its polarizability, or, depends upon the direction of the electric field relative to the molecule. Non-spherical rotors satisfy this condition. Therefore, linear and symmetric rotors are rotationally Raman active. 

---